Question

In a network as shown in the figure, the potential difference across the resistance $2\,R$ is (the cell has an emf of $E$ volts and has no internal resistance)

• A. $2E$
• B. $ \frac{4E}{7} $
• C. $ \frac{E}{7} $
• D. $ E $

Answer 1

Answer: (B)

Potential difference across resistors connected in parallel is same.
In the given circuit, resistors $4\, R$ and $2\, R$ are connected in parallel while resistance $R$ is connected in series to it. Hence, equivalent resistance is

$\frac{1}{R^{'}}=\frac{1}{4\, R}+\frac{1}{2 \,R}=\frac{6 R}{8 R^{2}}$
$R^{'}=\frac{8}{6} R=\frac{4}{3} R$
$\Rightarrow R^{''}=R+\frac{4}{3} R=\frac{7 R}{3}$
Given, emf is $E$ volts, therefore

$i=\frac{E}{R}=\frac{3 E}{7 R}$
Potential difference across $R$ is
$V=i r=\frac{3 E}{7 R} \times R=\frac{3 E}{7}$
Potential difference across $2 R$ is
$V^{'}=E-\frac{3 E}{7}=\frac{4 E}{7}$
Note : Potential drop across $4 \,R$ resistor is same as that across $2 \,R$, since both are connected in parallel.